insanely hard question

Question

It turns out that cos(36?) is half of the golden ratio. That is, cos(36?) =(1 + √5)/4
. As a result, we know that sin(36?) =(sqrt(10-2sqrt5)/4)
. Determine cos(9?) and sin(9?) without a calculator

Answer 1

Use the half-angle formulas: sin(θ/2) = √[(1-cos θ)/2] and cos(θ/2) = √[(1+cos θ)/2]

 

cos 18° = √[(1+cos 36°)/2] = √{[1+(1+√5)/4]/2} = √[(5+√5)/8] = √(10+2√5)/4

 

sin 9° = √[(1-cos 18°)/2] = √{[1-√(10+2√5) / 4]/2}

         = √{[4-√(10+2√5)]/8} = √[8-2√(10+2√5)]/4

 

cos 9° = √[(1+cos 18°)/2] = √{[1+√(10+2√5) / 4]/2}

          = √{[4+√(10+2√5)]/8} = √[8+2√(10+2√5)]/4